# Number Gossip

(Enter a number and I'll tell you everything you wanted to know about it but were afraid to ask.)

## Unique Properties of 866

- 866 is the smallest number n such that the sum of the proper divisors of n
^{2}, excluding 1, is a square
- 866 is the smallest number that together with the next 4 numbers have the property that they are not divisible by any of their nonzero digits

## Common Properties of 866

The number n is called an *apocalyptic power* if 2^{n} contains the consecutive digits 666 (in decimal).

A positive integer greater than 1 that is not prime is called *composite*.

Composite numbers are opposite to prime numbers.

The number n is *deficient* if the sum of all its positive divisors except itself is less than n.

Compare with perfect and abundant numbers.

A number is *even* if it is divisible by 2.

Numbers that are not even are odd. Compare with another pair -- evil and odious numbers.

The number n is *odious* if it has an odd number of 1's in its binary expansion.

Guess what evil numbers are.

A number is said to be *square-free* if its prime decomposition contains no repeated factors.

The next *Ulam* number is uniquely the sum of two earlier distinct Ulam numbers.